Convergence speed

The convergence speed of the genetic algorithm tends to increase with increasing population size. However, this is merely a general tendency, which interferes with the influence of the remaining adjustable parameters, and only for particular combinations of them becomes really apparent. 

Figure 2-1. Simple model to evaluate convergence speed in multipole expansion of the solvation energy...

In simple ergodic cases, the indices satisfy r = 2, = 1 but rj < 1 and 2, < 1 in the multiergodic cases, and the index 2, is easily defined from the scaling form of the mean PSD function—that is, (S(0) N)) = 0(N f). The frequency dependence of the indices seems to characterize the convergence speed of the power spectrum of resonant modes, but the details are still open when we treat the continuous-flow systems with many degrees of freedom. 

Note that the convergence speed of the extended trapezoid rule is quite slow. Thus, the method has to be used in this form only when the integral does not require a massive computation effort... 

If the substitution method converges, the convergence speed can be improved using the Aitken method (Buzzi-Ferraris and Manenti, 2010a). 

When the first derivative is zero, problems arise as the theoretical basis of the model is no longer valid. The convergence speed is no longer quadratic, for... 

The secant method has a convergence speed raised to the power of 1.618. It is slower than Newton s method, but the first derivative does not need to be evaluated. When the computational effort involved in evaluating the derivative is in the order of the computational time required to calculate the function, the secant method... 

Its main disadvantage is that its convergence speed is slower than the secant method. 

This device has the advantage of a better convergence speed, similar to the one of secant method. Many programs adopt this strategy as a basic algorithm (often combined with Bolzano s method to guarantee convergence even with complex and/or nonmonotone functions). 

This should not be considered as a convergence test, but merely useful information regarding the program s convergence speed. 

If there are only a few calculations for each one-dimensional search, the convergence speed tends to be linear. 

Newton s method (equations (3.57) and (3.58)) should be always used as the first method, if possible, to exploit its convergence speed whenever the matrix G is either reevaluated or updated. 

This method can be considered as an extension of the statistical approach to develop a comparative study of the behavior of different natural fibers. In this method, the density function of a quasi-stationary random process is estimated by means of an adaptive activation function neuron (FAN) endowed with a specific unsupervised learning theory with algorithms based on a neural network. The learning parameters could be chosen by carrying out several simulations. Here, one considers the values that provide a good trade-off between the convergence speed and the numerical stability of the algorithms [48]. 

A modification that can lead to a more significant improvement in convergence speed is... 

One possible modification of pure Newton s method is to rqtlace the search direction by H)BkI+Fi(ik)r k)> where is a positive parameter that produces a compromise between steepest descent (when is very large), and Newton s original method (> en Sk is zero). The aim is to achieve the robustness of steepest descent (applicable v en one starts ftir fi ( n the minimum) by an apprq>riately large choice of e k, and subsequently achieve the convergence speed of Newton s method (once one gets close to the minimum) by choosing a smaller value of 8k. Note that it is always necessary to choose 8k so that [s kl + F( ] is positive definite. 

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